96 Optimizing Optimization
The efficient surface is bounded on the upper left by the traditional mean –
variance efficient frontier, which is composed of efficient portfolios in dimensions
of expected return and standard deviation. The leftmost portfolio on the mean –
variance efficient frontier is the riskless asset. The right boundary of the efficient
surface is the mean – tracking error efficient frontier. It is composed of portfolios
that offer the highest expected return for varying levels of tracking error. The
leftmost portfolio on the mean – tracking error efficient frontier is the benchmark
portfolio because it has no tracking error. The efficient surface is bounded on the
bottom by combinations of the riskless asset and the benchmark portfolio. All
of the portfolios that lie on this surface are efficient in three dimensions. It does
not necessarily follow, however, that a three-dimensional efficient portfolio is
always efficient in any two dimensions. Consider, for example, the riskless asset.
Although it is on both the mean – variance efficient frontier and the efficient sur-
face, if it were plotted in dimensions of just expected return and tracking error,
it would appear very inefficient if the benchmark included high expected return
assets such as stocks and long-term bonds. This asset has a low expected return
compared to the benchmark and yet a high degree of tracking error.
Multigoal optimization will almost certainly yield an expected result that is
superior to constrained mean – variance optimization in the following sense. For
a given combination of expected return and standard deviation, it will produce a
portfolio with less tracking error. Or for a given combination of expected return
and tracking error, it will identify a portfolio with a lower standard deviation.
Or finally, for a given combination of standard deviation and tracking error, it
will find a portfolio with a higher expected return than a constrained mean –
variance optimization. Most of the portfolios identified by constrained mean –
variance optimization would lie beneath the efficient surface. In fact, multigoal
optimization would fail to improve upon a constrained mean – variance optimi-
zation only if the investor knew in advance what constraints were optimal. But,
of course, this knowledge could only come from multigoal optimization.
Figure 4.2 The efficient surface.